Properties of Exponents

When we read a number 34, 38, 4399, they are easy to read. Consider the number 294800000000—it is not so easy to read. 

But when we express the same value as 2948×10 8, it is quick and easy to say and understand. Properties of exponents are methods by which we denote the power of a number. They make it comparatively easy to express large numbers, which helps simplify multiplication and division and help solve problems easily.

Definition of exponents

Exponents are expressions comprising the repeated power of multiplication of the same number. It is also called raised to the power or power of a natural number. 

Let’s consider:  xn

Here,

x is the ‘base’

n is the ‘exponent’

Hence, xn is read as ‘x to the power of n’ (or) ‘x raised to n’.

Let us take an example.

• Consider a number 8 4.

• 8 is the base number.

• 4 is the power of the exponent.

• Now to evaluate the value of the exponent, the base is to be multiplied by the number of times of the power. 

• Without an exponent writing, the product will be difficult. The literal meaning of this is 8 X 8 X 8 X 8= 4096. If there were no exponents, we would not be able to express it in a short way.

• Thus, the number 4086 can be represented by 8 4. 

Other examples include:

3 3 = 3X3X3

4 6 = 4X4X4X4X4X4

Types of exponents

Based on the number in the power, exponents are of the following types:

• Positive exponent: It is simplified by multiplying the base to itself the number of times indicated by the exponent/power.

•  Negative exponent: It is simplified by placing 1 in the numerator and the base along with the exponent in the denominator of a fraction.

• Rational exponent: It is simplified with the denominator of the exponent being kept external to the root, while the base number is kept as root, with its power in the numerator.

• Zero exponent: It is already simple as it does necessarily have to be calculated as the exponent value is 0 to 1.

Properties of exponents 

To solve problems involving exponents, there are several properties that are considered major rules or laws that have to be followed while solving exponents. The seven properties of exponents are as follows.

• Law of product: am × an = am+n

• Law of quotient: am/an = am-n

• Law of zero exponent: a0 = 1

• Law of negative exponent: a-m = 1/am

• Law of power of a power: (am)n = amn

• Law of power of a product: (ab)m = ambm

• Law of power of a quotient: (a/b)m = am/bm

Law of product

The law of product is for any term with no 0,  am × an = am+n

Here, m and n = real numbers. 

For example, the simplification of

 6 6 X6 1= 6 6+1= 6 7.

This holds true for negative values of m and n, i.e. any integers.

An example is -6 -3X -6 -2 = -6 -3-2 = -6 -5

Law of quotient

The law of quotient holds true where  am/an = am-n

Here the non-zero term is A 

Integers – m and n.

For example, the value when 10 -5 is divided by 10 -2 is 

   10 -5/ 10 -2

= 10 -5 -(- 2 )

= 10 -5+2

= 10 -3

= 1/1000

Law of zero component

According to this law, the value is already simple as it does not even have to be calculated as the exponent value is 0 to 1.

Law of the negative component

This law tells us the number of times one must multiply the reciprocal of the base. 

An example would be if, given that a-n, it can be expanded as 1/an.

This means the reciprocal of a is 1/a which should be multiplied n times. 

When writing fractions using exponents, negative exponents are utilised.

Examples include 2 × 3-9, 7-3, 67-5, etc.

Law of power of a power

The base is a, and the power raised to the of the base  will give the product of the power,

(am)n = amn

Here, the non-zero term is a.

 Integers = m and n

For example, to express 8 3 as power, we use 2 as base.

= 2X2X2 = 8 = 2 3

Law of power of a product

According to the law, if the power is the same for two or different bases, then;

an bn = (ab)n

Here, the non-zero term is 0.

the integer is n.

Law of power of a quotient

This law states that the fraction of two different bases with the same power is represented as:

an/bn = (a/b)n

Here, the non-zero term is a and b.

The integer is m.

Conclusion 

Exponent properties can be used to convey the values of large numbers by using the exponent to represent the total number of times the base number must be multiplied with the same base. Exponents allow us to abbreviate something that would otherwise be really tedious to write. It has been widely used not only in mathematics but also to understand scientific scales like the Richter scale, and minute notations are used to write very large or very small numbers.